In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of

0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋.
4 2 4 4 2 4

a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
cos 3𝜋/4

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Understand that the problem involves finding the value of \( \cos \frac{3\pi}{4} \) using the unit circle, where each point on the circle corresponds to an angle \( t \) and has coordinates \( (x, y) = (\cos t, \sin t) \).

Locate the angle \( \frac{3\pi}{4} \) on the unit circle. Since the circle is divided into eight equal arcs, each arc corresponds to an angle of \( \frac{2\pi}{8} = \frac{\pi}{4} \). The angle \( \frac{3\pi}{4} \) is the third division from 0, moving counterclockwise.

Identify the coordinates of the point on the unit circle at \( t = \frac{3\pi}{4} \). This point lies in the second quadrant, where the x-coordinate (cosine) is negative and the y-coordinate (sine) is positive.

Recall the exact values for cosine and sine at multiples of \( \frac{\pi}{4} \). Specifically, \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \) and \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \).

Conclude that the value of \( \cos \frac{3\pi}{4} \) is the x-coordinate of the point on the unit circle at that angle, which is \( -\frac{\sqrt{2}}{2} \).

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Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, starting from the positive x-axis and moving counterclockwise. Understanding how angles correspond to points on the circle is essential for evaluating trigonometric functions like cosine and sine.

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Introduction to the Unit Circle

Coordinates on the Unit Circle

Each point on the unit circle corresponds to an angle t and has coordinates (x, y), where x = cos(t) and y = sin(t). These coordinates represent the cosine and sine values of the angle, respectively. Knowing how to read or find these coordinates allows direct evaluation of trigonometric functions.

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Introduction to the Unit Circle

Cosine Function and Its Geometric Interpretation

The cosine of an angle in the unit circle is the x-coordinate of the corresponding point. For angles like 3𝜋/4, which lie in the second quadrant, cosine values are negative. Recognizing the quadrant helps determine the sign and value of cosine without memorizing all values.

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